β€’ Necessary for computing summary statistics β€’ Sums of numerical values β€’ Denoted by the capital Greek letter sigma (Ξ£) 𝒏 𝑿 π’Š = π‘ΏπŸ + π‘ΏπŸ + π‘ΏπŸ‘ + β‹― + 𝑿𝒏 π’Š=𝟏 β€’ The summation of the sum (or difference) of two or more terms equals the sum (or difference) of the individual summations β€’ The summation of a constant, c, times a variable, X, equals the constant times the summation of the variable β€’ The summation of a constant, c, from i = 1 to n, equals the product of n and c. β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’ The subscript may be any letter, but the most common are i, j, and k. The lower limit of the summation may start with any number The lower limit of the summation is not necessarily a subscript The sum of the squared values of X is NOT equal to the square of the sum of X The summation of the square of (X+Y) is NOT equal to the summation of the sum of X2 and the sum of Y2 The sum of the product of X and Y is NOT equal to the product of the sum of values of X and the sum of values of Y The sum of the quotient of X and Y is NOT equal to the quotient of the sum of X and the sum of Y The sum of the square root of X is not equal to the square root of the sum of X β€’ Most common average β€’ Sum of all observed values divided by the number of observations β€’ For ungrouped and grouped data β€’ Population mean (ΞΌ) β€’ Sample mean ( 𝑋 ) β€’ Different formulas for grouped and ungrouped data β€’ β€’ β€’ β€’ β€’ Most common measure of central tendency Uses all observed values May or may not be an actual value in the data set May be computed for both grouped and ungrouped data sets Extreme observations affect the value of the mean β€’ The sum of the deviations of the observed values from the mean is zero. 𝒏 π‘Ώπ’Š βˆ’ 𝑿 π’Š=𝟏 = 𝟎 β€’ The sum of the squared deviations of the observed values from the mean is smallest. β€’ The Weighted Mean β€’ There are weights β€’ Values are not of equal importance β€’ e.g. GWA β€’ The Combined Mean β€’ Mean of several data sets β€’ The Trimmed Mean β€’ Less affected by extreme observations β€’ Order data β€’ Remove a certain percentage of the lower and upper ends β€’ Calculate arithmetic mean β€’ β€’ β€’ β€’ Middle value in an ordered set of observations Divides ordered set of observations into two equal parts Positional middle of an array Grouped and ungrouped data β€’ Extreme values affect the median less than the mean β€’ The median is used when β€’ We want the exact middle value of the distribution β€’ There are extreme observed values β€’ FDT has open-ended intervals β€’ β€’ β€’ β€’ Most frequent observed value in the data set Small data sets: inspection Large data sets: array or FDT Less popular β€’ β€’ β€’ β€’ β€’ β€’ Gives the most typical value of a set of observations Few low or high values do not easily affect the mode May not be unique and may not exist Several modes can exist Value of the mode is always in the data set May be used for both quantitative and qualitative data sets β€’ Data is symmetric and unimodal οƒ  all three measures may be used β€’ Data is asymmetric οƒ  use median or mode (if unique); trimmed mean β€’ Describe shape of data οƒ  use all three Ungrouped 𝟏 𝑿 = 𝒏 Grouped 𝒏 π‘Ώπ’Š π’Š=𝟏 𝑿= π’Œ π’Š=𝟏 π’‡π’Š π‘Ώπ’Š π’Œ π’Š=𝟏 π’‡π’Š Ungrouped β€’ n is odd 𝑀𝑑 = 𝑋 Grouped 𝑛 𝑀𝑑 = 𝐿𝐢𝐡𝑀𝑑 + 𝐢 𝑛 +1 2 β€’ n is even 𝑋 𝑀𝑑 = 𝑛 2 +𝑋 2 𝑛 +1 2 2 βˆ’< 𝐢𝐹𝑀𝑑 βˆ’1 𝑓𝑀𝑑 β€’ Grouped π‘€π‘œ = πΏπΆπ΅π‘€π‘œ π‘“π‘€π‘œ βˆ’ 𝑓1 +𝐢 2π‘“π‘€π‘œ βˆ’ 𝑓1 βˆ’ 𝑓2 Find the MEAN, MEDIAN, and MODE of the ff: 1. A sample survey in a certain province showed the number of underweight children under five years of age in each barangay: 3 5 6 4 7 8 6 9 10 4 6 7 5 8983455 2. Given the frequency distribution table of scores β€’ Measure of central tendency β€’ Measure of location β€’ Positional middle β€’ Percentiles β€’ Deciles β€’ Quartiles β€’ Divide ordered observations into 100 equal parts β€’ 99 percentiles; roughly 1 percent of observations in each group β€’ Interpretation: P1, the first percentile, is the value below which 1 percent of the ordered values fall. (ETC.) β€’ Ungrouped data β€’ Empirical Distribution Number with Averaging β€’ Weighted Average Estimate β€’ Grouped data β€’ Divide the ordered observations into 10 equal parts β€’ Each part has ten percent of the observations β€’ Divides observations into 4 equal parts β€’ Each part has 25 percent of the observations Q1 D1 P10 D2 P20 Q2 D3 P30 D4 P40 D5 P50 Q3 D6 P60 D7 P70 D8 P80 D9 P90 Empirical β€’ nk/100 is an integer 𝑋 π‘ƒπ‘˜ = π‘›π‘˜ 100 +𝑋 π‘›π‘˜ +1 100 2 β€’ nk/100 is not an integer π‘ƒπ‘˜ = 𝑋 π‘›π‘˜ +1 100 Weighted Average (𝑛 + 1)π‘˜ =𝑗+𝑔 100 π‘ƒπ‘˜ = 1 βˆ’ 𝑔 𝑋 𝑗 + 𝑔𝑋(𝑗 +1) π‘ƒπ‘˜ = πΏπΆπ΅π‘ƒπ‘˜ + 𝐢 π‘›π‘˜ βˆ’< 100 πΆπΉπ‘ƒπ‘˜ βˆ’1 π‘“π‘ƒπ‘˜ β€’ The number of incorrect answers on a true-false exam for a random sample of 20 students was recorded as follows: 2 β€’ and 2 find the 90th percentile, 7th decile, and 1st quartile β€’ Given the frequency distribution of scores of 200 students in an entrance exam in college, find the 95th percentile, 4th decile, and 3rd quartile. Scores Freq.